Vectors & 3D Geometry PYQ — Page 13

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k},\alpha >0$. If the projection of $\vec{a}\times \vec{b}$ on the vector $-\hat{i}+2\hat{j}-2\hat{k}$ is $30$, then $\alpha$ is equal to

The shortest distance between lines (x-1)/2=(y+1)/3=z/1 and (x+1)/5=(y-2)/1=z/(-1) is:

Let $\vec{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$. If $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to

Let $\vec{a}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. If $\vec{a}\times (2\hat{i}+\hat{k})=2\hat{i}-13\hat{j}-4\hat{k}$, then the projection of the vector $\vec{a}$ on the vector $2\hat{i}+2\hat{j}+\hat{k}$ is

Let $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a}\times \vec{b}=2\hat{i}-\hat{k}$ and $\vec{a}\cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is:

The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is

The direction cosines of the line joining points (1,2,3) and (4,6,3) are:

If the two lines ${l}_{1}:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and ${l}_{2}:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ are perpendicular, then an angle between the lines ${l}_{2}$ and ${l}_{3}:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonal of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute. $|\vec{a}|=1$ and $|\vec{a}.\vec{b}|=|\vec{a}\times \vec{b}|$. If $\vec{c}=2\sqrt{2}(\vec{a}\times \vec{b})-2\vec{b}$, then an angle between $\vec{b}$ and $\vec{c}$ is

Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors $(\vec{p}+\vec{q})$ and $(\vec{p}-\vec{q})$, and $|\vec{r}|=\sqrt{3}$, then $|\alpha |+|\beta |+|\gamma |$ is equal to

The angle between vectors a=i+j and b=j+k is:

If the projection of the vector $\hat{i}+2\hat{j}+\hat{k}$ on the sum of the two vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $-\lambda \hat{i}+2\hat{j}+3\hat{k}$ is $1$, then $\lambda$ is equal to _______.

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area (in sq. units) of triangle having vertices $(\alpha ,\beta ),(0,\beta )$ and $(0,0)$ is equal to

A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p+1$ and $\sqrt{10},$ then a value of $p$ is equal to:

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the value of $|(\vec{a}\times \vec{b})\times \vec{c}|$ is:

Let $a,b\in R.$ If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20,b,-a-9),$ then $|a+b|$ is equal to:

Let $\vec{c}$ be a vector perpendicular to the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+3\hat{k})=8$, then the value of $\vec{c}\cdot (\vec{a}\times \vec{b})$ is equal to

If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda )=(z-3),\lambda \in R$ is $\frac{1}{\sqrt{38}},$ then the integral value of $\lambda$ is equal to:

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}.$ Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}.$ If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is $19$ units, then $|2\vec{v}{|}^{2}$ is equal to _____.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{a}\cdot \vec{b}=1.$ If the length of projection vector of the vector $\vec{b}$ on the vector $\vec{a}\times \vec{c}$ is $l,$ then the value of $3{l}^{2}$ is equal to _____.

A hall has a square floor of dimension $10m\times 10m$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is ${\mathrm{cos}}^{-1}\frac{1}{5},$ then the height of the hall (in meters) is: 

Let $\vec{a}=\hat{i}+5\hat{j}+\alpha \hat{k},\vec{b}=\hat{i}+3\hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\vec{b}\times \vec{c}|=5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}.$ Then the greatest amongst the values of $|\vec{a}{|}^{2}$ is ________.

Let $a,b$ and $c$ be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then $c$ is equal to:

If $\vec{a}$ and $\vec{b}$ are unit vectors and $(\vec{a}+3\vec{b})$ is perpendicular to $(7\vec{a}-5\vec{b})$ and $(\vec{a}-4\vec{b})$ is perpendicular to $(7\vec{a}-2\vec{b}),$ then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is _________.